Suppose X is a scheme over a field K, and write Xbar for the basechange of X to Kbar, so that as usual we have an exact sequence.
Now there may be no section from G_K back to . But certainly X has a rational point over some finite extension L/K, which means that there is definitely a section from the finite-index subgroup G_L to . This is so easy that I can’t help wondering: is there a way to see the existence of such a “virtual section” from group theory alone? My intuition is to say no. But I just thought I’d mention it, while we’re puzzling anabelianly.